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Kagi brings its 'small web' of a human-only internet to mobile devices | TechCrunch
(techcrunch.com)
This is a most excellent place for technology news and articles.
They talk about the name in the initial announcement back in 2023 where they link to many blogs discussing the topic.
https://blog.kagi.com/small-web
The term is a bit broader than those protocols and Kagi is far from the first to use it; it certainly isn't a "hijack" as if it was the name if another project or something. 'Small Web' isn't them claiming to own the concept of the small web, or that it's somehow only accessible through them... It's just a feature; search, as a curated product they offer and maintain.
It's just what they named the lens, because it's a lens for the 'small web' as they defined it; like the other lenses. They aren't hijacking the word 'academia' by having an academia lens...
https://help.kagi.com/kagi/features/lenses.html#default-lenses
Sure, maybe they could slap 'Kagi' in front of 'Small Web' just to be sure, but I doubt anyone will confuse the concept of noncommercial small websites with a paid service...
Also
Neat fact: you can trace the roots of trademark law back like 7000 years ...
Because I thought you were obviously wrong about the 7000 years thing, here's a history of trademarks by some guy named Olivier Pierre:
I've gotten so used to think of trademarks as registered trademarks, but it makes sense that it has existed much longer in the literal sense. The earliest known law however dates back little more than 4000 years, and there's nothing about trademarks there, so I think it's fair to say trademark law is a lot more modern. :)
Sorry for being entirely off-topic.